Critical points/ Abs. max/min values

Hey guys so I need help on two questions that I'm sort of stuck on.

1. Find the critical numbers of the given function: f(x) = 2x^{3}+x^{2}+2x

So I got f'(x) = 6x^{2}+2x+2

Factored out the 2(3x^{2}+x+1)

I can't factor this out anymore, so is the answer there are no critical numbers?

2. Find the absolute max and min values of f on the given interval.

√ = square root

f(t) = ^{3}√t * (8-t), [0,8]

So I used product rule on this one and got

f'(t) = 3/2√t * (8-t) - ^{3}√t = 0

3(8-t)-3√t = 0

So, I'm uncertain about the critical numbers here as well are they 8 and 0?

Re: Critical points/ Abs. max/min values

1) If you sketch a cubic equation you will see that it has no maximum or minimum.

2) There's a mistake in your differentiation of t^{1/3}

Re: Critical points/ Abs. max/min values

Quote:

Originally Posted by

**Shakarri** 1) If you sketch a cubic equation you will see that it has no maximum or minimum.

2) There's a mistake in your differentiation of t^{1/3}

Yeah I just noticed the derivative was wrong for number 2.

For 1 do I just leave my answer as it is? and write no critical points then?

Re: Critical points/ Abs. max/min values

For the first one the reason is that the quadratic 3x^2+x+1 does not have real roots. Its discriminant is negative so it has imaginary roots and hence no critical points.

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Re: Critical points/ Abs. max/min values

Re: Critical points/ Abs. max/min values

Quote:

Originally Posted by

**ibdutt**

Quote:

Originally Posted by

**ibdutt** For the first one the reason is that the quadratic 3x^2+x+1 does not have real roots. Its discriminant is negative so it has imaginary roots and hence no critical points.

Have I told you how much I love you?

Haha, thanks a lot for both of the questions. You are the best!

Re: Critical points/ Abs. max/min values

So the only critical point for question 2 will be 0, and when looking for absolute max/ min, with the intervals, the f(x) would give 0 for both the numbers in the interval. So, is it possible to have a abs. max/min on the same point?

Re: Critical points/ Abs. max/min values